The fundamental matrix of the Hilbert problem in transport theory - Rajamäki, Markku (Transport theory and statistical physics, 1973)
A method of solving the homogeneous Hilbert problem encountered in the two-media case of the one-dimensional transport equation with degenerate scattering kernels is developed. The existence of the corresponding fundamental matrix is proved when the fundamental matrices for both the full-range problems exist and the elements of these matrices and of their inverses belong to a class L p , p < 2, and the zeroes of the dispersion functions are of integer order. Matrices related to the fundamental matrices satisfy a Fredholm equation whose free term involves unknown polynomial elements. Means of constructing the fundamental matrix from certain solutions of this equation are investigated. Sufficient supplementary conditions are presented in the form of an algebraic equation. Since the kernel is simple in form and its norm approaches zero with the cross sections of both media approaching each other, the Fredholm equation is very suitable for computation.